A new super PAC largely funded by two billionaire Republican donors plans to run ads attacking Hillary Clinton in battleground states, hoping that will buttress the party’s chances of retaining its Senate majority and winning the White House.

Casino magnate Sheldon Adelson is expected to spend at least $5 million on the effort, while TD Ameritrade Holding Corp. founder Joe Ricketts and his family will contribute at least $1 million. The super PAC’s organizers say other donors have pledged support and that their goal is to...